Problem: Rei is barricading a door to stop a horde of zombies. She stacks boxes of books on a table in front of the door. Each box weighs $30$ kilograms, and the table with $8$ boxes on top weighs a total of $310$ kilograms. The total weight $W$ of the barricade in kilograms is a function of $x$, the number of boxes Rei stacks on the table. Write the function's formula. $W=$
Solution: The weight of each box is constant, so we're dealing with a linear relationship. We could write the desired formula in slope-intercept form: $W= mx+ b$. In this form, $ m$ gives us the slope of the graph of the function and $ b$ gives us the $y$ -intercept. Our goal is to find the values of $ m$ and $ b$ and substitute them into this formula. We know that each box increases the weight of the barricade by $30$ kilograms, so the slope $ m$ is ${30}$, and our function looks like $W={30}x+ b$. We also know that with $8$ boxes on the table, the total weight of the barricade is $310$ kilograms, which means that when $x=8$, $W=310$. We can substitute this into the formula of the function to find $ b$ : $\begin{aligned}{30}\cdot8+ b&=310\\\\ 240+ b&=310\\\\ b&={70}\end{aligned}$ This means the weight of the table is $70$ kilograms. Since $ m = {30}$ and $ b = {70}$, the desired formula is: $W={30}x+{70}$